Simplify the Boolean expression (X + Y + XY)(X + Z) to its minimal sum form using absorption and distributive properties.
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AXZ + Y
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BX + Y + Z
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CX + YZ
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DXY + YZ
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ENone of the above
Answer
Correct Answer: X + YZ
Explanation
Introduction / Context:Boolean simplification reduces logic cost, delay, and power. This problem reinforces core identities such as absorption and distributivity to reach a minimal or near-minimal expression suitable for implementation.
Given Data / Assumptions:
- Expression: (X + Y + XY)(X + Z).
- Variables X, Y, Z are Boolean (0/1).
- Goal: simplify to a compact sum-of-terms form.
Concept / Approach:Use absorption: A + AB = A. Therefore, Y + XY = Y. Then use the distributive identity: (X + A)(X + B) = X + AB. Apply these stepwise to collapse the expression.
Step-by-Step Solution:
Start: (X + Y + XY)(X + Z).Apply absorption on (Y + XY): X + (Y + XY) = X + Y.Now expression becomes: (X + Y)(X + Z).Use distributive form: (X + Y)(X + Z) = X + YZ.Result: X + YZ.Verification / Alternative check:Construct a quick truth check: when X=1, result is 1; when X=0, result reduces to YZ. This matches the product form behavior, confirming the simplification.
Why Other Options Are Wrong:
- XZ + Y: does not match cases where X=0 and Y=1, Z=0 (gives 1 instead of YZ=0).
- X + Y + Z: over-approximates (too many 1s).
- XY + YZ: misses cases where X alone should drive 1.
- None of the above: incorrect because X + YZ is correct.
Common Pitfalls:Forgetting absorption (Y + XY = Y), or expanding fully then miscombining minterms leading to longer forms.
Final Answer:X + YZ